1. **Problem:** Lines r and s are parallel. We need to find the slope of each line using slope triangles and conclude about the slopes of parallel lines. 2. **Formula:** Slope $m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}$. 3. Since lines r and s are parallel, their slope triangles have the same shape and size ratios, so their slopes are equal. 4. Let the slope triangle on line r have rise $\Delta y_1$ and run $\Delta x_1$, so slope $m_r = \frac{\Delta y_1}{\Delta x_1}$. 5. Similarly, slope on line s is $m_s = \frac{\Delta y_2}{\Delta x_2}$. 6. Because the triangles are similar, $\frac{\Delta y_1}{\Delta x_1} = \frac{\Delta y_2}{\Delta x_2}$, so $m_r = m_s$. 7. **Conclusion:** Parallel lines have equal slopes. --- 1. **Problem:** Lines m and n are perpendicular. Find slopes using slope triangles and conclude about slopes of perpendicular lines. 2. **Formula:** Slope $m = \frac{\text{rise}}{\text{run}}$. 3. For line m, slope $m_m = \frac{\Delta y}{\Delta x}$. 4. For line n, slope $m_n = \frac{\Delta y'}{\Delta x'}$. 5. Since lines are perpendicular, their slopes satisfy $m_m \times m_n = -1$. 6. This means $m_n = -\frac{1}{m_m}$. 7. **Conclusion:** Slopes of perpendicular lines are negative reciprocals. --- 1. **Problem:** How are slope triangles, corresponding sides, ratios, and rise/run related? 2. Slope triangles are right triangles drawn on a line to visualize rise and run. 3. Corresponding sides in similar slope triangles maintain the same ratio. 4. The ratio of rise to run ($\frac{\text{rise}}{\text{run}}$) is the slope. 5. Therefore, slope triangles help us understand slope as a ratio of vertical change to horizontal change. --- 1. **Problem:** A student found slopes 4 and $\frac{8}{2}$ on the same line and concluded slopes differ. 2. Calculate $\frac{8}{2} = 4$. 3. Both slopes are equal to 4. 4. Since slope is constant on a line, the student’s conclusion is incorrect. 5. **Correction:** The slope is the same at all points on a line. --- 1. **Problem:** True or false? If one vertex of a slope triangle is at (0,0), the other vertex on the line represents the simplified unit rate or slope. 2. This is true because the slope triangle from (0,0) to another point $(x,y)$ on the line has rise $y$ and run $x$. 3. Simplifying $\frac{y}{x}$ gives the unit rate or slope. --- 1. **Problem:** Does placement of slope triangles matter when finding slope? 2. No, because slope is constant along a line. 3. Slope triangles anywhere on the line are similar, so their rise/run ratios are equal. 4. Therefore, placement does not affect slope calculation.